So bear with me if you can. I will try to type my understanding (which is bad) to this subject (regarding simultaneity), what's exactly the question and why it has came up.
Why
Reading some papers on numerical relativity (Any search of "Numerical Relativity" or "Magnetically Arrested Disk" will result in papers, not posting certain papers for obvious reason. They study a closed system, and the observer, which is us humans, are at inf distance) they refer to "simulation time".
Let's take Kerr Spacetime geometry for example. They relate that simulation time by multiplying it to $\dfrac{GM}{c^3}$, which is just the "unit time" to go from simulation to "real" time, and that's the proper time that passes for a ZAMO observer on infinity (us). Basically its the same quantity multiplied by unity ($\delta \tau = \delta t$)
That got me thinking because I'm not really that experienced with General Relativity (have studied some books, mostly recently, but still my intuition is not good) how can we correlate the coordinate time to an observer time for events that happened so far away (not referring to information travel speed) so easily.
Me, trying to understand it
So what I will try and elaborate here, are things that I have mostly read on 3+1 Formalism in General Relativity which is actually a free e-book from Éric Gourgoulhon and I only refer to it because one can check Chapter 2,3,4 and find these things for himself (hopefully you won't need for this question as I'll try to explain them to my understanding).
If we consider Hypersurfaces of constant coordinate $t$, then, at each point, we can define a unit vector which is prepandicular to that Hypersurface. An example of this is the following picture:
Hypersurfaces of a spacetime, where $t$ is the coordinate time and $\mathbf{n}$ is the ZAMO/Eulerian timeline at a certain point. There are infinite points, so there are infinite ZAMO observers.
Now, I will post a "part" of text that comments on simultaneity of that observer:
Since $\mathbf{n}$ is a unit timelike vector, it can be regarded as the 4-velocity of some observer. We call such observer an Eulerian observer. The worldlines of the Eulerian observers are thus orthogonal to the hypersurfaces $\Sigma _t$. Physically, this means that the hypersurface $\Sigma _t$ is locally the set of events that are simultaneous from the point of view of the Eulerian observer, according to Einstein-Poincaré simultaneity convention
From this mark, only the local events that are orthogonal to the ZAMO time-like vector are simultaneous.
So from the "Why" paragraph before, our purpose is to correlate the coordinate time $t$ with the proper time $\tau$ of a ZAMO/Eulerian Observer.
Again, quoting from the book (Italic are comments), we got this:
Let us consider two close events $p$ and $p'$ on the worldline of some Eulerian observer (Meaning that both of these points are on the "line" characterized by n vector).
Let $t$ be the “coordinate time” of the event $p$ and $t + \delta t$ $(\delta t > 0)$ that of $p'$ , in the sense that $p \in \Sigma _t$ and $p' \in \Sigma _{t+\delta t}$. Then $p' = p + \delta t \mathbf{m}$ (consider $\mathbf{m}$ the vector that travels from one hypersurface to the other, and as one can intuitively guess, it's linear to $\mathbf{n}$. No need to over-analyze this I feel like).
The proper time $\delta \tau$ between the events $p$ and $p'$, as measured the Eulerian observer, is given by the metric length of the vector linking $p$ and $p'$:
$$\delta \tau = \sqrt{\mathbf{-g(m,m)}} \delta t = N\delta t$$ (Where $N$ is the lapse function $g^{00} = -\tfrac{1}{N^2}$)
Notice that the lapse function $N$ goes to unity $N_{r \rightarrow \infty} = 1$, so from this equation, the time parameter is actually the proper time, for events on the worldline n.
Of course, the "area" which is perpendicular to the $\mathbf{n}$ as we go to $\infty$ (meaning that the point on Hypersurface of the $\mathbf{n}$ is at infinite) is "the whole" local space. As we can see from this gif, the events on the same parameter $t$ are indeed simultaneous.
The question
Since in bibliography they translate the parameter time $t$ immediately to the proper time $\tau$ of our observer, the "right" answer should be that all events on hypersurface $\Sigma _t$ are simultaneous to the observer in $\infty$.
But how can we prove it? Since from the second quote, there is no immediate result regarding 2 events at "curved" spacetime in relation to the Lorentzian Spacetime in $\infty$.
Really thank you for even reading to this point. Wasn't sure if I should just "throw" the question or try to include some information for it to be more complete. Since in the future, one might have similar questions, I decided to do the last. Sorry for the long text :)
Any input is helpful!
EDIT:
To be a bit more clear, from the above 3+1 General Relativity theory, there is an observer who is called Eulerian/ZAMO and his 4-velocity is always perpendicular to the $\Sigma _t$ Hyper-surfaces.
These observers are unique at each point of the Hyper-surface. For these observers, we can consider what is "simultaneous".
The relation $\delta \tau = N \delta t$ shows how much proper time has passed for these events, and the gif shows what can be considered "simultaneous" for an observer. (Only one can be stattic with respect to the Kerr coordinates though, so one of those 2 observers, would not be considered ZAMO).
So the final question:
Can we prove that the hypersurface of $\Sigma _t$ is the surface of the events that happen at the same proper time of an Eulerian Observer/ ZAMO at $\infty$ ?
